Bayes’ Theorem
If we say that a whole event contains 100 basic events and event A contains 40 basic events. Based on classical probability theory, we can say p(A) = 40/100 = 0.4
If we keep on saying that B has the events of 40 also and both A and B has the event number of 20. Now we observe event B happens, what is the probability that the happened event is A?
Yes, it will be (A and B)/B = 20/40 = 0.5
Then you were told, C also happens, the events within the C category is 40 also, and the events of A and B and C is 10, A and C is 20, B and C is 15. what is the probability of A under this new observation?
Since both B and C happens. The possibility of A happen will be P(A and (B and C)) /P(B and C) = P(A and B and C)/ P(B and C) = 10/15 = 2/3.
Of course, if P(A and C) is zero, and if C happens, we can get the conclusion that A is impossible.
This whole process demonstrates the power of Bayes. More related evidence that we have, the better conclusion that we can get. Of course, if the evidence is totally not related to what we want to predict, it will be meanless. For example, if we have another whole event. Evidence from the other world will not be helpful to this world.
Reference
